Abstract
Neural solvers for partial differential equations (PDEs) have great potential, yet their practicality is currently limited by their generalizability. PDEs evolve over broad scales and exhibit diverse behaviors; predicting these phenomena will require learning representations across a wide variety of inputs, which may encompass different coefficients, geometries, or equations. As a step towards generalizable PDE modeling, we adapt masked pretraining for PDEs. Through self-supervised learning across PDEs, masked autoencoders can learn useful latent representations for downstream tasks. In particular, masked pretraining can improve coefficient regression and timestepping performance of neural solvers on unseen equations. We hope that masked pretraining can emerge as a unifying method across large, unlabeled, and heterogeneous datasets to learn latent physics at scale.
Overview
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The paper presents a new method using masked autoencoders and self-supervised learning for solving partial differential equations (PDEs), aimed at simulating complex phenomena more efficiently.
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Masked autoencoding enhances model adaptability by obscuring parts of input data during training, forcing the model to learn richer representations applicable to a variety of PDE-related tasks.
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Experiments show that this approach yields better results in coefficient regression and PDE timestepping tasks than traditional supervised methods, notably improving predictions of PDE dynamics.
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The study highlights the potential of masked autoencoders in scientific computing, indicating their versatility and promise for future applications in higher-dimensional PDEs and tasks like super-resolution.
Masked Autoencoders as Effective Tools for Learning Partial Differential Equations
Introduction
The exploration of neural solutions for partial differential equations (PDEs) represents a significant intersection of machine learning and mathematical modeling, pivotal for simulating complex phenomena such as fluid dynamics, material deformation, and climate systems. The traditional computational methods for solving these equations, while effective, often struggle with adapting to varying conditions or new equations without extensive retraining or data recollection. This paper introduces a novel approach to enhance the adaptability and efficiency of neural solvers for PDEs through masked autoencoders, leveraging self-supervised learning to attain useful latent representations for an array of downstream tasks, including coefficient regression and timestepping for unseen equations.
Methodology
The proposed method applies masked autoencoding, a technique wherein parts of the input data are intentionally obscured during training to encourage the model to learn richer internal representations. This method, successful in domains such as natural language processing and image processing, is adapted to handle the complexity of PDEs. The authors employ Transformer-based architectures, specifically designed for 1D and 2D PDE representations, to encode visible patches of spatiotemporal PDE data, with the decoder reconstructing the full data from these partial inputs. The inclusion of Lie point symmetry data augmentations enhances the diversity of the training set, augmenting the model's ability to generalize across a broader spectrum of PDE-related tasks.
Experiments and Results
The study's experiments span a variety of PDEs—ranging from the 1D KdV-Burgers equation to 2D Heat, Advection, and Burgers equations—to evaluate the model's capability in coefficient regression and timestepping tasks. These tasks aim to assess the model's utility in predicting PDE dynamics, both within a learned representation space and in generating future states of PDE systems. The results show marked improvements over traditional supervised methods, particularly when fine-tuning the models on specific PDE tasks. Key numerical outcomes include:
- Coefficient Regression: In both 1D and 2D scenarios, pretrained models significantly outperform their randomly initialized counterparts in predicting equation coefficients. For instance, in 1D PDE regression, a fine-tuned pretrained model achieved a mean squared error (MSE) that is an order of magnitude lower than that of a supervised baseline.
- PDE Timestepping: Using autoencoder embeddings to inform neural PDE solvers (e.g., Fourier Neural Operators) led to substantially reduced errors in autoregressively predicting future PDE states. This confirms that the latent representations learned through masked pretraining effectively encapsulate salient features of the PDE dynamics.
Implications and Future Directions
The findings underscore the considerable potential of masked autoencoders in the realm of PDE learning, offering a pathway toward models that can efficiently adapt to novel equations or conditions without the need for retraining from scratch. This accelerates the deployment of neural solvers in various scientific and engineering applications, from weather forecasting to the design of metamaterials. The versatility and adaptation shown by these models reaffirm the value of self-supervised learning in extracting meaningful patterns from complex datasets.
Looking ahead, the exploration could extend to even more complex or higher-dimensional PDEs, incorporating advanced attention mechanisms to manage the increased data complexity. Furthermore, applying the pretrained models to tasks like super-resolution could open new avenues in enhancing the accuracy and usability of numerical simulations across disciplines. The adaptability of masked autoencoders to varied architectures and tasks, combined with their capacity to utilize large, unlabeled datasets, positions them as a promising tool for foundational models in the scientific computation domain.
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